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Common preference, non-consequential features, and collective decision making

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Abstract

This paper examines an extended framework of Arrovian social choice theory. We consider two classes of values: consequential values and non-consequential values. Each individual has a comprehensive preference based on the two. Non-consequential values are assumed to be homogeneous among individuals. It is shown that a social ordering function satisfying Arrovian conditions must be non-consequential: a social comprehensive preference gives unequivocal priority to non-consequential values. We clarify the role of common preferences over non-consequential features.

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Notes

  1. Perfit (1984) discusses the implications of consequentialism.

  2. For the argument on non-consequentialism, see Scheffler (1982), Pettit (2000), and Suzumura (1999). For criticisms of welfarism, see Rawls (1971) and Dworkin (1977).

  3. See discussions between Fleurbaey et al. (2003) and Kaplow and Shavell (2004). See also Blackorby et al. (2005).

  4. As we will see later, it does not have a positive role in characterizing “non-consequential” social orderings.

  5. A binary relation \(R\) is complete if and only if, for all \((x, \theta ),(y, \theta ') \in X \times \varTheta \), \((x,\theta ) R(y,\theta ')\) or \( (y,\theta ')R (x,\theta )\). A binary relation \(R\) is transitive if and only if, for all \((x, \theta ),(y, \theta '),(z, \theta '') \in X \times \varTheta \), \([(x,\theta ) R(y,\theta ') \text{ and } (y,\theta ')R (z,\theta '')] \Rightarrow (x,\theta )R (z,\theta '')\).

  6. Then, \((x,\theta )I(y,\theta ') {\Leftrightarrow } [(x,\theta )R(y,\theta ') \text{ and } (y,\theta ')R(x,\theta )]\); \(xPy {\Leftrightarrow } [(x,\theta )R(y,\theta ') \text{ and } \lnot (y,\theta ')R(x,\theta )]\).

  7. Cato (2012) provides a systematic treatment of social choice without the Pareto principle.

  8. Cato (2014) discusses the informational-efficiency aspect of the independence of irrelevant alternatives.

  9. Sen (1970) and Hammond (1976) examine another class of extended frameworks of Arrovian social choice. They incorporate information about interpersonal comparisons. Each individual has a preference over the Cartesian product of \(X\) (the set of alternatives) and \(N\) (the set of individuals). Then, \((x,j)R_i(y,k)\) means that individual \(i\) weakly prefers individual \(j\)’s position at \(x\) to individual \(k\)’s position at \(y\).

  10. Sen (1986) discusses the informational implications of IIA and emphasizes its Kantian aspect.

  11. Proposition 1 is closely related to Theorem 4 of Suzumura and Xu (2004). However, the domain conditions between our work and Suzumura and Xu’s are different. Essentially, our preference domain is a superdomain of that of Suzumura and Xu (2004). See Sect. 5 below.

  12. Note that social decision making on social outcomes is implemented in a dictatorial manner under the constructed social ordering function in the proof of Theorem 1.

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Acknowledgments

An earlier version of this paper was circulated under the title “The role of common morality in social choice.” I am grateful to Katsuhito Iwai, Kazuya Kamiya, Tomohiko Kawamori, Masahiro Okuno-Fujiwara, Masayuki Otaki, Toyotaka Sakai, Dan Sasaki, Kotaro Suzumura, Naoki Yoshihara, two anonymous referees, an associate editor, and Atila Abdulkadiroglu of this journal for helpful comments and suggestions. This paper was financially supported by Grant-in-Aid for Young Scientists (Start-up; B) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology.

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Correspondence to Susumu Cato.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

‘If’. We show that if there exists at least one non-consequentialist, then there exists a social ordering function \(f\) satisfying PP, IIA, and ND.

Step 1. By assumption, there exists a non-consequentialist \(i \in N\). Choose two individuals \(k,l \in N\) with \(k \ne l\). Let \(\theta ^* \in \varTheta \). Consider the following social ordering function \(f\): for all \((x,\theta ), (y,\theta ') \in X \times \varTheta \),

$$\begin{aligned} \theta J \theta '&\Rightarrow (x,\theta ) P (y,\theta ')\\ \theta =\theta '=\theta ^*&\Rightarrow [(x,\theta ) R_l (y,\theta ') \Leftrightarrow (x,\theta ) R (y,\theta ')]\\ \theta =\theta '\ne \theta ^*&\Rightarrow [(x,\theta ) R_k (y,\theta ') \Leftrightarrow (x,\theta ) R (y,\theta ')]. \end{aligned}$$

Step 2. By constriction, \(f\) satisfies PP, IIA, and ND. \(R=f(\mathbf{R})\) is reflexive and complete. Thus, it suffices to show that \(R\) is transitive, i.e., \(\forall (x,\theta ),(y,\theta '),(z,\theta '') \in X \times \varTheta , (x,\theta ) R (y,\theta ') \wedge (y,\theta ') R (z,\theta '') \Rightarrow (x,\theta ) R (z,\theta '')\).

Then, to check transitivity, we consider the four possibilities that are mutually exclusive and exhaustive.

  1. (1)

    \(\theta J \theta '\) and \(\theta ' J \theta ''\). By transitivity of \(J\), we have \(\theta J \theta ''\). This implies \((x,\theta ) R (z,\theta '')\).

  2. (2)

    \(\theta J \theta '\) and \(\theta ' = \theta ''\). By transitivity of \(J\), we have \(\theta J \theta ''\). This implies \((x,\theta ) R (z,\theta '')\).

  3. (3)

    \(\theta =\theta '\) and \(\theta ' J \theta ''\). By transitivity of \(J\), we have \(\theta J \theta ''\). This implies \((x,\theta ) R (z,\theta '')\).

  4. (4)

    \(\theta = \theta '\) and \(\theta ' =\theta ''\).

    1. (4-a)

      \(\theta = \theta '= \theta ''=\theta ^*\). In this case, \((x,\theta ) R (y,\theta ')\) \(\wedge \) \((y,\theta ') R (z,\theta '')\) implies that \((x,\theta ) R_l (y,\theta ')\) \(\wedge \) \((y,\theta ') R_l (z,\theta '')\). By transitivity of \(R\), we have \((x,\theta ) R_l (z,\theta '')\). By the construction of the social ordering function, \((x,\theta ) R (z,\theta '')\).

    2. (4-b)

      \(\theta = \theta ' =\theta ''\ne \theta ^*\). In this case, \((x,\theta ) R (y,\theta ')\) \(\wedge \) \((y,\theta ') R (z,\theta '')\) implies that \((x,\theta ) R_k (y,\theta ')\) \(\wedge \) \((y,\theta ') R_k (z,\theta '')\). By transitivity of \(R\), we have \((x,\theta ) R_k (z,\theta '')\).

Therefore, \(R\) is transitive.

‘Only if’. We show that if there exists no non-consequentialist, then there exists no social ordering function \(f\) satisfying PP, IIA, and ND. Since every individual is a consequentialist, for all \(i \in N\),

$$\begin{aligned} \forall (x,\theta ),(y,\theta ') \in X \times \varTheta : x \succ _i y&\Rightarrow (x,\theta ) P_i (y,\theta ') \text{ and } x \sim _i y\nonumber \\&\Rightarrow [(x,\theta ) R_i (y, \theta ') \Leftrightarrow \theta J \theta ']. \end{aligned}$$
(2)

First we prove the following claim:

(Claim) There exists \(d \in N\) such that \((x,\theta ) P_d (y,\theta ') \Rightarrow (x,\theta ) P (y,\theta ')\)

for all \((x,\theta ),(y,\theta ') \in X \times \varTheta \) with \(x \ne y\).

Consider a triple \((x,\theta ),(y,\theta '),(z,\theta '') \in X \times \varTheta \) such that \(x,y,z\) are all distinct. Note that individual’s extended preference is not restricted over this triple. Since \(\# \{(x,\theta ),(y,\theta '),(z,\theta '')\}=3\), Arrow’s impossibility theorem can be applied. Hence, there exists a local dictator \(d\) over \(\{(x,\theta ),(y,\theta '),(z,\theta '')\}\). Take \((a,\gamma )\) and \((b,\gamma ')\) such that \(a \ne b\) and \(a,b \in X \setminus \{x,y,z \}\). Now we show that \(d\) is a local dictator over \(\{(a,\gamma ),(b,\gamma ')\}\). Consider a triple \((a,\gamma ),(y,\theta '),(z,\theta '')\). We can apply Arrow’s impossibility theorem to the triple, and thus, there exists a local dictator over \(\{(a,\gamma ),(y,\theta '),(z,\theta '')\}\). The local dictator over \(\{(a,\gamma ),(y,\theta '),(z,\theta '')\}\) must be individual \(d\). Similarly, \(d\) is a local dictator over \(\{(a,\gamma ),(b,\gamma '),(z,\theta '')\}\). Therefore, \(d\) is a local dictator over \(\{(a,\gamma ),(b,\gamma ')\}\). Thus, the claim is proved.

Next, we show that for all \((x,\theta ),(y,\theta ') \in X \times \varTheta \), if \(x=y\), then \((x,\theta ) P_d (y,\theta ') \Rightarrow (x,\theta ) P (y,\theta ')\). Take any \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(x=y\). Since \(D_d \in {\mathcal {D}}_J\), \((x,\theta ) P_d (y,\theta ')\) if and only if \(\theta J \theta '\). Since every individual is a consequentialist, \(x =y\) and \(\theta J \theta '\) imply that \((x,\theta ) P_i (y,\theta ')\) for all \(i \in N\). By PP, \((x,\theta )P(y,\theta ')\). Therefore, \((x,\theta ) P_d (y,\theta ') \Rightarrow (x,\theta ) P (y,\theta ')\). Combining with the first claim, \(d\) is the universal dictator over \(X \times \varTheta \). \(\square \)

Proof of Lemma 1

For each \(i \in N\), for all \(x,y \in X\) and all \(\theta , \theta ' \in \varTheta \),

$$\begin{aligned} \Big [ \theta J \theta ' \Rightarrow (x,\theta ) P_i (y,\theta ') \Big ] \text{ and } \Big [ \theta = \theta ' \Rightarrow [(x,\theta ) R_i (y,\theta ') \Leftrightarrow x \succsim _i y] \Big ]. \end{aligned}$$

Thus, \(\theta J \theta '\) implies \((x,\theta ) P_i (y,\theta ')\) for all \(i \in N\). Since \(f\) satisfies PP, we have \((x,\theta ) P (y,\theta ')\). Then, \(\theta J \theta ' \Rightarrow (x,\theta ) P (y,\theta ')\). \(\square \)

Proof of Theorem 1

Suppose that a social ordering function \(f\) satisfies PP, IIA, and ND. Let \(\mathcal {W}\) and \(\mathcal {J}\) denote the set of consequentialists and the set of non-consequentialists, respectively. By assumptions, \(\mathcal {W} \cup \mathcal {J} = N\) and \(\mathcal {J} \ne \emptyset \). Then, there exists \(\langle (\succsim _i)_{i \in N}, J\rangle \) such that for each \(i \in \mathcal {W}\), for all \(x,y \in X\) and all \(\theta ,\theta ' \in \varTheta \),

$$\begin{aligned}&x \succ _i y \Rightarrow (x,\theta ) P_i (y,\theta '); \\&x \sim _i y \Rightarrow [(x,\theta ) P_i (y,\theta ') \Leftrightarrow \theta J \theta ' \text{ and } (x,\theta ) I_i (y,\theta ') \Leftrightarrow \theta = \theta ']. \end{aligned}$$

and for each \(i \in \mathcal {J}\), for all \(x,y \in X\) and all \(\theta , \theta ' \in \varTheta \),

$$\begin{aligned} \theta J \theta '&\Rightarrow (x,\theta ) P_i (y,\theta '); \\ \theta = \theta '&\Rightarrow [(x,\theta ) R_i (y,\theta ') \Leftrightarrow x \succsim _i y]. \end{aligned}$$

Lemma 1 implies that if \(\mathcal {W} = \emptyset \), the \(f\) is non-consequential. Then, throughout the rest of the proof, we assume that \(\mathcal {W} \ne \emptyset \).

First, we show that there exists a local dictator over \(\{(x, \theta ): x \in X \}\).

Step 1. For each \(\theta \in \varTheta \), there exists \(d(\theta ) \in N\) such that for all \(\mathbf{R} \in {\mathcal {D}}\), \((x,\theta ) P_{d(\theta )} (y,\theta ) \Rightarrow (x,\theta ) P (y,\theta )\) for all \(x,y \in X\).

Take any \(\theta ^* \in \varTheta \). Consider preference profiles restricted on \(\{(x, \theta ) \in X \times \varTheta :\theta = \theta ^* \}\). By our supposition on the preference domain, each individual preference ordering \(R_i\) corresponds to \(\succsim _i\) under \(\{(x, \theta ) \in X \times \varTheta :\theta = \theta ^* \}\). Since \(\succsim _i\) is not restricted and \(\# \{(x, \theta ) \in X \times \varTheta :\theta = \theta ^* \} \ge 3\), we can apply Arrow’s impossibility theorem to this subset. Hence, there exists \(d(\theta ^*) \in N\) such that for all \(\mathbf{R} \in {\mathcal {D}}\), \((x,\theta ^*) P_i (y,\theta ^*) \Rightarrow (x,\theta ^*) P (y,\theta ^*)\) for all \(x,y \in X\). \(\Box \)

We call individual \(d(\theta )\) a local dictator over circumstance \(\theta \). Next, we show the following result.

Step 2. Suppose that there exist \(\mathbf{R} \in {\mathcal {D}}\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\). Then, \(d(\theta )=d(\theta ')\).

By way of contradiction, suppose that there exist \(\mathbf{R} \in {\mathcal {D}}\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\), and \(d(\theta ) \ne d(\theta ')\). Let \(\mathbf{R}' \in {\mathcal {D}}\) be such that

$$\begin{aligned} \mathbf{R}'|\{(x,\theta ),(y,\theta ')\}&=\mathbf{R}|\{(x,\theta ),(y,\theta ')\},\\ (x,\theta )&P'_{d(\theta )} (z,\theta ),\\ (z,\theta ')&P'_{d(\theta ')} (y,\theta '). \end{aligned}$$

Since \(d(\theta )\) and \(d(\theta ')\) are distinct, the above profile is admissible. Since \(\mathbf{R}'|\{(x,\theta ),(y,\theta ')\}=\mathbf{R}|\{(x,\theta ),(y,\theta ')\}\), IIA implies that \((y,\theta ') R' (x,\theta )\) where \(R'=f( \mathbf{R' })\). Since \(d(\theta )\) is a local dictator over \(\theta \), \((x,\theta ) P'_{d(\theta )} (z,\theta )\) implies that \((x,\theta ) P' (z,\theta )\). Similarly, we have \((z,\theta ') P' (y,\theta ')\). Since \((y,\theta ')R(x,\theta )\), \((x,\theta ) P' (z,\theta )\) and \((z,\theta ') P' (y,\theta ')\), we obtain

$$\begin{aligned} (z, \theta ') P' (z, \theta ). \end{aligned}$$

Since \(\mathbf{R}' \in {\mathcal {D}}_J\), \((z,\theta ) P'_i (z,\theta ')\) for all \(i \in N\). By PP, we have

$$\begin{aligned}(z,\theta ) P' (z,\theta '). \end{aligned}$$

This is a contradiction. Hence, \(d(\theta ) = d(\theta ')\). \(\square \)

Step 3. If there exist distinct \(\theta ,\theta ' \in \varTheta \) such that \(d(\theta ) \ne d(\theta ')\), then \(f(\mathbf{R})\) is J-priori for all \(\mathbf{R} \in {\mathcal {D}}\).

By way of contradiction, suppose that there exist \(\mathbf{R} \in {\mathcal {D}}\) and \((x,\theta _1),(y,\theta _2) \in X \times \varTheta \) such that \(\theta _1 J \theta _2\) and \((y,\theta _2) R (x,\theta _1)\) where \(R=f(\mathbf{R})\). From Step 2, \(d(\theta _1) = d(\theta _2)\). Since there exist distinct \(\theta ,\theta ' \in \varTheta \) such that \(d(\theta ) \ne d(\theta ')\), there exists \(\theta _3 \in \varTheta \) such that \(d(\theta _3) \ne d(\theta _1)= d(\theta _2)\).

Let \(\mathbf{R}' \in {\mathcal {D}}\) be such that

$$\begin{aligned} \mathbf{R}'|\{(x,\theta _1),(y,\theta _2)\}&=\mathbf{R}|\{(x,\theta _1),(y,\theta _2)\},\\ (z,\theta _2) P'_{d(\theta _2)} (y,\theta _2)&\quad \text{ and } \quad (z,\theta _2)P'_{d(\theta _2)} (x,\theta _1). \end{aligned}$$

Then, IIA implies that \((y,\theta _2)R'(x,\theta _1)\). Since \(d(\theta _2)\) is a local dictator over \(\theta _2\), it follows that \((z,\theta _2) P' (y,\theta _2)\). Transitivity implies that \((z,\theta _2) P'_{d(\theta _2)} (x,\theta _1)\). The ranking of \((z,\theta _2)\) and \((x, \theta _1)\) is not specified for individuals except \(d(\theta _2)\), and thus, IIA implies that \((z,\theta _2) P_{d(\theta _2)} (x,\theta _1) \Rightarrow (z,\theta _2) P (x,\theta _1)\) for all \(\mathbf{R} \in {\mathcal {D}}\).

Let \(\mathbf{R}'' \in {\mathcal {D}}\) be such that

$$\begin{aligned} (z,\theta _2)&P''_{d(\theta _2)} (x,\theta _1),\\ (x,\theta _3)&P''_{d(\theta _3)} (z,\theta _3),\\ \theta _1 J'' \theta _3&\quad \text{ and } \quad \theta _3 J'' \theta _2. \end{aligned}$$

It is admissible that \(\theta _1 J' \theta _3\) and \(\theta _3 J' \theta _2\). By the above argument, \((z,\theta _2) P''_{d(\theta _2)} (x,\theta _1) \Rightarrow (z,\theta _2) P'' (x,\theta _1)\). Since \(\mathbf{R}'' \in {\mathcal {D}}_J\), \((x,\theta _1) P''_i (x,\theta _3)\) for all \(i \in N\). Then, PP implies \((x,\theta _1) P'' (x,\theta _3)\). Similarly, we have \((z,\theta _3) P'' (z,\theta _2)\). On the other hand, \(d(\theta _3) \in N\) is a local dictator over \(\theta _3\), and thus, \((x,\theta _3) P''_{d(\theta _3)} (z,\theta _3) \Rightarrow (x,\theta _3) P'' (z,\theta _3)\). Then, we summarize the social preference as follows:

$$\begin{aligned} (z,\theta _2) P'' (x,\theta _1),\\ (x,\theta _1) P'' (x,\theta _3), \\ (x,\theta _3) P'' (y,\theta _3), \\ (y,\theta _3) P'' (z,\theta _3),\\ (z,\theta _3) P'' (z,\theta _2), \end{aligned}$$

where \(R''=f(\mathbf{R}'')\). This contradicts transitivity. Hence, \(f(\mathbf{R})\) is J-priori for every profile \(\mathbf{R}\). \(\square \)

Now, we consider the case where \(d = d(\theta )= d(\theta ')\) for all \(\theta ,\theta ' \in \varTheta \). In other words, there exists \(d \in N\) such that for all \(\theta \in \varTheta \), \((x, \theta ) P_{d} (y, \theta ) \Rightarrow (x, \theta ) P (y, \theta )\) for all \(x, y \in X \). Either (i) \(d \in \mathcal {W}\) or (ii) \(d \in \mathcal {J}\). We first consider case (i).

Step 4. If there exists \(d \in \mathcal {W}\) such that for all \(\theta \in \varTheta \), \((x, \theta ) P_{d} (y, \theta ) \Rightarrow (x, \theta ) P (y, \theta )\) for all \(x, y \in X \), then \(f(\mathbf{R})\) is J-priori for all \(\mathbf{R} \in {\mathcal {D}}\).

On the contrary, suppose that there exist \(\mathbf{R} \in {\mathcal {D}}\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\).

(a) First, we show that for all \(z \in X\), \((z,\theta ') P_d (x,\theta ) \Rightarrow (z,\theta ') P (x,\theta )\). Let \(\mathbf{R}' \in {\mathcal {D}}\) be such that

$$\begin{aligned} \mathbf{R}'|\{(x,\theta ),(y,\theta ')\}=\mathbf{R}|\{(x,\theta ),(y,\theta ')\},\\ (z,\theta ') P'_d (x,\theta )\quad \text{ and } \quad (z,\theta ') P'_d (y,\theta '). \end{aligned}$$

Since \(d\) is a consequentialist, \(\mathbf{R}'\) is possible. IIA implies that \((y,\theta ') R' (x,\theta )\). Since \(d\) is a local dictator over \(\theta '\), we have \((z,\theta ') P' (y,\theta ')\). Since \((z,\theta ') P' (y,\theta ')\) and \( (y,\theta ') R' (x,\theta )\), we have \((z,\theta ') P' (x, \theta )\) by transitivity. The ranking of \((z,\theta ')\) and \((x, \theta )\) is not specified for individuals except \(d\), and thus, IIA implies that \((z,\theta ') P_d (x,\theta ) \Rightarrow (z,\theta ') P (x,\theta )\) for all \(\mathbf{R} \in {\mathcal {D}}\).

(b) Next, we show that \((z,\theta ') P_d (w,\theta ) \Rightarrow (z,\theta ') P (w,\theta )\) for all \(z,w \in X\). Let \(\mathbf{R}'' \in {\mathcal {D}}\) be such that

$$\begin{aligned} (z,\theta ') P''_d (x,\theta ), (x,\theta ) P''_d (w,\theta ) ,\quad \text{ and } \quad (z,\theta ') P''_d (w,\theta ). \end{aligned}$$

It follows from (a) that \((z,\theta ') P''_d (x,\theta ) \Rightarrow (z,\theta ') P'' (x,\theta )\). Since individual \(d\) is a local dictator over \(\theta \), we have \((x,\theta ) P'' (w,\theta )\). Since \((z,\theta ') P'' (x,\theta ) \) and \( (x,\theta ) P'' (w,\theta )\), \((z,\theta ') P'' (w,\theta )\). The ranking of \((z,\theta ')\) and \((w, \theta )\) is not specified for individuals except \(d\), and thus, IIA implies that for all \(\mathbf{R} \in {\mathcal {D}}\), \((z,\theta ') P_d (w,\theta ) \Rightarrow (z,\theta ') P (w,\theta )\) for all \(z,w \in X\).

(c) Finally, we show that \(d\) is the universal dictator, i.e., \((a, \bar{\theta }) P_d (b, \hat{\theta }) \Rightarrow (a, \bar{\theta }) P (b, \hat{\theta })\) for all \((a, \bar{\theta }) ,(b, \hat{\theta }) \in X \times \varTheta \). Let \(\mathbf{R}''' \in {\mathcal {D}}\) be such that

$$\begin{aligned} (a, \bar{\theta })&P'''_d (b, \hat{\theta }), \\ \theta J''' \hat{\theta }&\text{ and } \bar{\theta } J''' \theta '. \end{aligned}$$

Since \(R'''_i \in {\mathcal {D}}_J\) for all \(i \in N\), \((a, \bar{\theta }) P'''_i (a,\theta ')\) and \((b, \theta ) P'''_i (b, \hat{\theta })\) for all \(i \in N\). By PP, \((a, \bar{\theta }) P''' (a,\theta ')\) and \((b, \theta ) P''' (b, \hat{\theta })\). Since \((x, \theta ') P_d (y,\theta ) \Rightarrow (x,\theta ') P (y,\theta )\) for all \(x,y \in X\) by (b), we have \((a,\theta ') P''' (b,\theta )\). By transitivity, \((a, \bar{\theta }) P''' (b, \hat{\theta })\). The ranking of \((a, \bar{\theta })\) and \((b, \hat{\theta })\) is not specified for individuals except \(d\), and thus, IIA implies that \(d \in \mathcal {C}\) is the universal dictator. This is a contradiction. \(\Box \)

Next, we consider case (ii).

Step 5. If there exists \(d \in \mathcal {J}\) such that for all \(\theta \in \varTheta \), \((x, \theta ) P_{d} (y, \theta ) \Rightarrow (x, \theta ) P (y, \theta )\) for all \(x, y \in X \), then there exists no social ordering function \(f\) satisfying PP, IIA and ND.

If, for all \(\mathbf{R} \in {\mathcal {D}}\), \(\theta J \theta ' \Rightarrow (x,\theta ) R (y,\theta ')\) for all \((x,\theta ), (y,\theta ') \in X \times \varTheta \), then \(d \in \mathcal {J}\) is the universal dictator. This is inconsistent with ND. Hence, there exist \(\mathbf{R} \in {\mathcal {D}}\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\). Let \(\mathbf{R}' \in {\mathcal {D}}\) be such that

$$\begin{aligned} \mathbf{R}'|\{(x,\theta ),(y,\theta ')\}&=\mathbf{R}|\{(x,\theta ),(y,\theta ')\},\\ (x,\theta ) P'_d (y,\theta )&\text{ and } (x,\theta ') P'_d (y,\theta '), \\&\theta J' \theta '. \end{aligned}$$

Since \(d\) is a non-consequentialist, \(\mathbf{R}'\) is possible. By IIA, \((y,\theta ')R'(x,\theta )\) where \(R'=f(\mathbf{R'})\). Moreover, since \(d\) is a local dictator over \(\theta '\), \((x,\theta ') P'_d (y,\theta ') \Rightarrow (x,\theta ') P' (y,\theta ')\). By transitivity, \((x,\theta ') P' (x,\theta )\). Since \((x,\theta ) P'_i (x,\theta ')\) for all \(i \in N\), PP implies that \((x,\theta ) P' (x,\theta ')\). This is a contradiction. \(\square \)

From Steps 1–5, if \(f\) satisfies PP, IIA, and ND, then \(f(\mathbf{R})\) is J-priori for all \(\mathbf{R} \in {\mathcal {D}}\). \(\blacksquare \)

Proof of Theorem 2

Step 1. For each \(\theta \in \varTheta \), there exists \(d(\theta ) \in N\) such that \((x,\theta ) P_{d(\theta )} (y,\theta ) \Rightarrow (x,\theta ) P (y,\theta )\) for all \(x,y \in X\).

Take any \(\theta ^* \in \varTheta \). Note that CIIA implies the following: for all \(\mathbf{R} = (R_1,R_2,\cdots ,R_n)\), \(\mathbf{R}' = (R'_1,R'_2,\cdots ,R'_n) \in {\mathcal {D}} \), and for all \((x,\theta ^*),(y,\theta ^*)\) \(\in X \times \varTheta \), if

$$\begin{aligned} \mathbf{R}|\{(x,\theta ^*),(y,\theta ^*)\}=\mathbf{R}|\{(x,\theta ^*),(y,\theta ^*) \}, \end{aligned}$$

then \([ (x,\theta ^*) R (y,\theta ^*)\) \(\Leftrightarrow \) \((x,\theta ^*) R' (y,\theta ^*) ]\) and \([ (y,\theta ^*) R (x,\theta ^*)\) \(\Leftrightarrow \) \((y,\theta ^*) R' (x,\theta ^*) ]\). Then, CIIA can be applied to pairs over \(\{(x, \theta ) \in X \times \varTheta :\theta = \theta ^* \}\). In the same manner as in Step 1 of the proof of Theorem 1, we can apply Arrow’s impossibility theorem to this subset. Hence, the claim is proved. \(\square \)

Step 2. Suppose that there exist \(\mathbf{R} \in {\mathcal {D}}_J\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\). Then, \(d(\theta )=d(\theta ')\).

By way of contradiction, suppose that there exist \(\mathbf{R} \in {\mathcal {D}}_J\) and \((x,\theta ),(y,\theta ') \in X \times \varTheta \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\), and \(d(\theta ) \ne d(\theta ')\). Fix \(z \notin \{ x,y \}\). Let \(\mathbf{R}' \in {\mathcal {D}}_J\) be such that

$$\begin{aligned} \mathbf{R}'|\varGamma&=\mathbf{R}|\varGamma ,\\ (x,\theta )&P'_{d(\theta )} (z,\theta ) ,\\ (z,\theta ')&P'_{d(\theta ')} (y,\theta '), \end{aligned}$$

where \(\varGamma =\{ x,y \} \times \{ \theta ,\theta ' \}\). Since \(d(\theta )\) and \(d(\theta ')\) are distinct, the above profile is admissible. CIIA implies that \((y,\theta ') R' (x ,\theta )\). In the same manner as in Step 2 of the proof of Theorem 1, we obtain a contradiction, and thus, \(d(\theta ) = d(\theta ')\). \(\Box \)

Now we prove Theorem 2. Suppose, on the contrary, that there exist \(\mathbf{R} \in {\mathcal {D}}_J\) and \((x,\theta _1),(y,\theta _2) \in X \times \varTheta \) such that \(\theta _1 J \theta _2\) and \((y,\theta _2) R (x,\theta _1)\) where \(R=f(\mathbf{R})\). From Step 1, there exists a local dictator \(d(\theta )\) over each circumstance \(\theta \in \varTheta \); from Step 2, \(d(\theta _1) = d(\theta _2)\). By NCD, there exists \(\theta _3 \in \varTheta \) such that \(d(\theta _3) \ne d(\theta _1)\). Fix \(z \notin \{ x,y \}\). Let \(\mathbf{R}' \in {\mathcal {D}}_J\) be such that

$$\begin{aligned} \mathbf{R}'|\varGamma&=\mathbf{R}|\varGamma ,\\ (x,\theta _3)&P'_{d(\theta _3)} (z,\theta _3)\\ (z,\theta _2)&P'_{d(\theta _1)} (y,\theta _2)\\ \theta _1 J' \theta _3&\text{ and } \theta _3 J' \theta _2, \end{aligned}$$

where \(\varGamma =\{ x,y \} \times \{ \theta ,\theta ' \}\). It is admissible that \(\theta _1 J' \theta _3\) and \(\theta _3 J' \theta _2\). CIIA implies that \((y,\theta _2) R' (x, \theta _1)\). Since \(\mathbf{R}' \in {\mathcal {D}}_J\), \((x,\theta _1) P'_i (x,\theta _3)\) and \((z,\theta _2) P'_i (z,\theta _3)\) for all \(i \in N\). By CPP, \((x,\theta _1) P' (x,\theta _3)\) and \((z,\theta _3) P' (z,\theta _2)\). On the other hand, \(d(\theta _3) \in N\) is the local dictator over the circumstance \(\theta _3\), and thus, \((x,\theta _3) P'_{d(\theta _3)} (z,\theta _3) \Rightarrow (x,\theta _3) P' (z,\theta _3)\). Similarly, \((z,\theta _2) P'_{d(\theta _1)} (y,\theta _2) \Rightarrow (z,\theta _2) P' (y,\theta _2)\). Note that

$$\begin{aligned} (y,\theta _2) R' (x, \theta _1) P' (x,\theta _3) P' (z,\theta _3)P' (z,\theta _2)P' (y,\theta _2). \end{aligned}$$

This contradicts transitivity. \(\blacksquare \)

Proof of Theorem 3

Step 1. For each \(\theta \in \varTheta \), there exists \(d(\theta ) \in N\) such that \((x,\theta ) P_{d(\theta )} (y,\theta ) \Rightarrow (x,\theta ) P (y,\theta )\) for all \(x,y \in \{ a \in X: (a,\theta ) \in \varPsi \}\).

Take any \(\theta ^* \in \varTheta \). Since \(\varPsi \) is connected, \(\# \{ a \in X: (a,\theta ) \in \varPsi \}\ge 3\). Then, Arrow’s impossibility theorem can be applied in the same manner as in Step 1 of the proof of Theorem 2. \(\square \)

Step 2. Suppose that there exist \(\mathbf{R} \in {\mathcal {D}}^{\varPsi }_J\) and \((x,\theta ),(y,\theta ') \in \varPsi \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\) where \(R=f(\mathbf{R})\). Then, \(d(\theta )=d({\theta '})\).

By way of contradiction, suppose that there exist \(\mathbf{R} \in {\mathcal {D}}^{\varPsi }_J\) and \((x,\theta ),(y,\theta ') \in \varPsi \) such that \(\theta J \theta '\) and \((y,\theta ') R (x,\theta )\), and \(d(\theta ) \ne d({\theta '})\). Since \(\varPsi \) is connected, there exists \(z \in X\) such that \((z,\theta ), (z,\theta ') \in \varPsi \). Let \(\mathbf{R}' \in {\mathcal {D}}^{\varPsi }_J\) be such that

$$\begin{aligned} \mathbf{R}'|(\varGamma \cap \varPsi )&=\mathbf{R}|(\varGamma \cap \varPsi ),\\ (x,\theta )&P'_{d(\theta )} (z,\theta ) ,\\ (z,\theta ')&P'_{d(\theta ')} (y,\theta '), \end{aligned}$$

where \(\varGamma = \{ x, y \} \times \{ \theta ,\theta ' \}\). In the same manner as in Step 2 of the proof of Theorem 2, we obtain a contradiction. \(\Box \)

We prove Theorem 3. Suppose, on the contrary, that there exist \(\mathbf{R} \in {\mathcal {D}}^{\varPsi }_J\) and \((x,\theta _1),(y,\theta _2) \in X \times \varTheta \) such that \(\theta _1 J \theta _2\) and \((y,\theta _2) R (x,\theta _1)\) where \(R=f(\mathbf{R})\). Steps 1 and 2 imply that there exists a local dictator over each circumstance \(\theta \in \varTheta \) and \(d({\theta _1}) = d(\theta _2)\). NCD implies that there exists \(\theta _3 \in \varTheta \) where \(d(\theta _3) \ne d(\theta _1)\).

Consider the case where \((x,\theta _3) \in \varPsi \). Since \(\varPsi \) is connected, there exists \(z \notin \{ x,y \}\) such that \((z,\theta _2),(z,\theta _3) \in \varPsi \). Let \(\mathbf{R}' \in {\mathcal {D}}_J\) be such that

$$\begin{aligned}&\mathbf{R}'|(\varGamma \cap \varPsi )=\mathbf{R}|(\varGamma \cap \varPsi )\\&(x,\theta _3) P'_{d(\theta _3)} (z,\theta _3)\\&(z,\theta _2) P'_{d(\theta _1)} (y,\theta _3)\\&\theta _1 J' \theta _3 \text{ and } \theta _3 J' \theta _2, \end{aligned}$$

where \(\varGamma = \{ x, y \} \times \{ \theta _1 ,\theta _2 \}\). CIIA implies that \((y,\theta _2) R' (x, \theta _1)\). Since \(\mathbf{R}' \in {\mathcal {D}}_J\), \((x,\theta _1) P'_i (x,\theta _3)\) and \((x,\theta _2) P'_i (x,\theta _3)\) for all \(i \in N\). By CPP, \((x,\theta _1) P' (x,\theta _3)\) and \((z,\theta _3) P' (z,\theta _2)\). Moreover, \((x,\theta _3) P'_{d( \theta _3 )} (z,\theta _3) \Rightarrow (x,\theta _3) P' (z,\theta _3)\) and \((z,\theta _2) P'_{d(\theta _1)} (y,\theta _3) \Rightarrow (z,\theta _2) P' (y,\theta _3)\). Note that

$$\begin{aligned} (y,\theta _2) R' (x, \theta _1) P' (x,\theta _3) P' (z,\theta _3) P' (z,\theta _2) P' (y,\theta _3). \end{aligned}$$

This contradicts transitivity.

Consider the case where \((x,\theta _3) \notin \varPsi \). Since \(\varPsi \) is connected, there exist distinct \(z,w \notin \{ x,y \}\) such that \((z,\theta _1),(z,\theta _3) \in \varPsi \) and \((w,\theta _2),(w,\theta _3) \in \varPsi \). Let \(\mathbf{R}'' \in {\mathcal {D}}_J^{\varPsi }\) be such that

$$\begin{aligned}&\mathbf{R}''|(\varGamma \cap \varPsi )=\mathbf{R}|(\varGamma \cap \varPsi )\\&(z,\theta _3) P''_{d({\theta _3})} (w,\theta _3)\\&(x,\theta _1) P''_{d({\theta _1})} (z,\theta _1)\\&(w,\theta _2) P''_{d({\theta _1})} (y,\theta _2)\\&\theta _1 J'' \theta _3 \text{ and } \theta _3 J'' \theta _2, \end{aligned}$$

where \(\varGamma = \{ x, y \} \times \{ \theta _1 ,\theta _2 \}\). CIIA implies that \((y,\theta _2) R'' (x, \theta _1)\). CPP implies that \((z,\theta _1) P'' (z,\theta _3)\) and \((w,\theta _3) P'' (w,\theta _2)\). Moreover, \((z,\theta _3) P''_{d( \theta _3 )} (w,\theta _3) \Rightarrow (z,\theta _3) P'' (w,\theta _3)\), \((x,\theta _1) P''_{d({\theta _1})} (z,\theta _1) \Rightarrow (x,\theta _1) P'' (z,\theta _1)\), and \((w,\theta _2) P''_{d({\theta _1})} (y,\theta _2) \Rightarrow (w,\theta _2) P'' (y,\theta _2)\). Note that

$$\begin{aligned} (y,\theta _2) R'' (x, \theta _1)P'' (z,\theta _1)P'' (z,\theta _3)P'' (w,\theta _3)P'' (w,\theta _2)P'' (y,\theta _2). \end{aligned}$$

This contradicts transitivity. \(\blacksquare \)

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Cato, S. Common preference, non-consequential features, and collective decision making. Rev Econ Design 18, 265–287 (2014). https://doi.org/10.1007/s10058-014-0164-3

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